Single Linkage Clustering and Continuum Percolation

Suppose f is a probability density function in d dimensions, d ≥ 2. A single linkage a-cluster on a sample of size n from the density f is a connected component of the union of balls of volume a, centred at the sample points. Let λc be the percolation threshold above which a d-dimensional Poisson process of rate λ has an unbounded 1-cluster. We show that for large n, the "big" single linkage (λc/(hn))-clusters can be used to detect population clusters, i.e., maximal connected sets of the form {x : f(x) ≥ h}. Here, a big cluster is one that contains a positive fraction of the sample points.